FINDING THE POINT OF DISCONTINUITY WORKSHEET

Find any points of discontinuity for each rational function.

Problem 1 :

y = (x + 3)/(x – 4) (x + 3)

Solution

Problem 2 :

y = (x - 2)/(x² – 4)

Solution

Problem 3 :

y = (x - 3) (x + 1)/(x – 2)

Solution

Problem 4 :

y = 3x(x + 2)/x(x + 2)

Solution

Problem 5 :

y = 2/(x + 1)

Solution

Problem 6 :

y = 4x/(x³ – 9x)

Solution

Problem 7 :

Check the function

g(x) = (x² + 7x + 10)/(x - 3)(x + 2)

for discontinuities. Conduct appropriate tests to determine if asymptotes exist at the discontinuity values. State the equations of any asymptotes and the domain of g(x)

Solution

Problem 8 :

Match the equation of each rational function with the most appropriate graph. Explain your reasoning.

a) y = (x + 4) / (x² - 3x - 4)

b) y = (x + 4) / (x² + 5x + 4)

c) y = (x² + 4x)/(x + 4)

Solution

Problem 9 :

Write the equation for each graphed rational function

Solution

Problem 10 :

Write the equation for each graphed rational function.

Solution

Find the points of discontinuity.  Classify each  point as a vertical asymptote or a hole.

Problem 1 :

y = x/(x² – 9)

Solution

Problem 2 :

y = (3x² – 1)/x³

Solution

Problem 3 :

y = (6x² + 3)/(x – 1)

Solution

Problem 4 :

y = (5x³ – 4)/(x² + 4x – 5)

Solution

Problem 5 :

y = 7x/(x³ + 1)

Solution

Problem 6 :

y = (12x⁴ + 10x – 3)/3x⁴

Solution

Problem 7 :

y = (12x + 24)/(x² + 2x)

Solution

Problem 8 :

y = (x² – 1)/(x² + 3x + 2)

Solution

Problem 9 :

y = (x² – 1)/(x² - 2x - 3)

Solution

State whether or not each of the following functions is continuous.

• If not, state where the discontinuity occurs and whether or not it is removable.
• Is the discontinuity is an asymptote, a hole or jump ?
• If an asymptote, what is its equation ?

Problem 1 :

f(x) = x/(x² + 1)

Solution

Problem 2 :

f(x) = x/(2x² - x - 1)

Solution

Problem 3 :

f(x) = (2x + 3)/(x² - x - 6)

Solution

Problem 4 :

f(x) = (x - 4)/(x² - 16)

Solution

Problem 5 :

Solution

Problem 6 :

Solution

Problem 7 :

Solution

Problem 8 :

Solution

Problem 9 :

Find the value of a if the function is continuous.

Solution

Problem 10 :

Find the value of a if the function is continuous.

Solution

Problem 12 :

The graph of the function 𝑓(𝑥) is shown below :

Which of the following statements is true about 𝑓?

I. 𝑓 is undefined at 𝑥 = 1.

II. 𝑓 is defined but not continuous at 𝑥 = 2.

III. 𝑓 is defined and continuous at 𝑥 = 3.

(A) Only I     (B) Only II      (C) I and II     (D) I and III

(E) None of the statements are true.

Solution